| L(s) = 1 | + 5-s + 2·11-s − 4·13-s − 17-s − 5·19-s + 9·23-s − 4·25-s + 6·29-s − 8·31-s − 11·37-s + 4·41-s + 43-s − 2·47-s − 6·53-s + 2·55-s + 9·59-s + 6·61-s − 4·65-s + 7·67-s + 71-s + 4·73-s + 8·79-s − 4·83-s − 85-s + 15·89-s − 5·95-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s − 1.10·13-s − 0.242·17-s − 1.14·19-s + 1.87·23-s − 4/5·25-s + 1.11·29-s − 1.43·31-s − 1.80·37-s + 0.624·41-s + 0.152·43-s − 0.291·47-s − 0.824·53-s + 0.269·55-s + 1.17·59-s + 0.768·61-s − 0.496·65-s + 0.855·67-s + 0.118·71-s + 0.468·73-s + 0.900·79-s − 0.439·83-s − 0.108·85-s + 1.58·89-s − 0.512·95-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.943371172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.943371172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.43915778369711, −13.13384846029815, −12.56849009511589, −12.21579239418533, −11.68988074947645, −10.99554319911965, −10.74805424837693, −10.17903224513083, −9.469890463782734, −9.357118777072784, −8.643383438141836, −8.297987666230866, −7.525100036382230, −6.891416272023502, −6.794971840561408, −6.073695010203921, −5.388487994560044, −4.997509564221383, −4.467541479635363, −3.743163038832715, −3.256646628648890, −2.373799094811841, −2.082597362085184, −1.279376434766795, −0.4268473750957154,
0.4268473750957154, 1.279376434766795, 2.082597362085184, 2.373799094811841, 3.256646628648890, 3.743163038832715, 4.467541479635363, 4.997509564221383, 5.388487994560044, 6.073695010203921, 6.794971840561408, 6.891416272023502, 7.525100036382230, 8.297987666230866, 8.643383438141836, 9.357118777072784, 9.469890463782734, 10.17903224513083, 10.74805424837693, 10.99554319911965, 11.68988074947645, 12.21579239418533, 12.56849009511589, 13.13384846029815, 13.43915778369711
